1...
Introduction To
The Ideas Of
The Quantum Theory
The Modern Viewpoint in Science
Roy Lisker
#306 Liberty Commons
8 Liberty Street
Middletown , CT 06457
www.fermentmagazine.org
CHAPTER I
THE EXISTENT AND THE KNOWABLE
The physics of the 20th century has forced philosophy beyond its traditional boundaries. Our notions about the universe have been reduced to uncomforting prejudices: among these one includes mass, energy, space, time, the criteria of objectivity, the relationship of the observer to the observed, and the role of experiment in the evolution of a scientific theory.
Few things have proven more insubstantial than substance itself. No dividing line now separates matter from energy ; particle and wave appear as the two faces of the same dual reality. Substance , once believed co-extensive with matter , has been supplanted by the field , a kind of localization of the potential of coming into being. Time itself , once considered so inherent to nature as to be in no need of description or analysis, now enters physics as a dependent variable only , linked to gravitation, relative velocity, and space.
It is therefore all the more surprising that our control over the environment is far more confident than ever in the past. From the incurable uncertainties of quantum theory come the flawless accuracy of the laser, the high speed computer, the transistor microchip. [1] The abolition of an uncontingent time and space by the special and general theories of relativity has not resulted in chaos in our daily lives. To the contrary , they have given us the power to peer billions of light-years into space, thereby taking in the full extent of the heavens, and to witness , as if unfolding before our very eyes, the split second of the creation of the universe , the dawn of time .
This new amplitude of vision, this grand perspective on encompassing reality is surrounded , much as a solid oak thickly coated in lichens and moss, by a cornucopia of ingenious world systems. Practical, applied and theoretical science have given rise to a richness of speculative science. The liberation of the intellect has catalyzed the liberation of the imagination: Big Bangs [2] , Big Crunches [3] , Cosmic Inflation [4] , Strings [5] , Quantum Foam [6] , Wormholes [7] , Non-Locality [8], Hidden Variables [9] , Anthropism [10] , Many Worlds [11] , etc .....
The bankruptcy of conventional ways of thinking about matter was apparent to most physicists in the period just before the turn of the century. The watershed years of the conceptual revolution, or in Thomas Kuhn’s [12] vocabulary , the paradigm shift , occurred between 1900 , when Max Planck discovered his quantum of action , and 1905 , when Einstein put the finishing touches on the special theory of relativity. These radical theories emerged in response to contradictions within and between:
(I) The Maxwell-Lorentz theory of electromagnetism, and
(II) The theory of heat, (Thermodynamics) .
Electromagnetism : It was recognized that one could no longer give mechanistic interpretations , ( in the sense of Newtonian mechanics ) , to basic electromagnetic phenomena . In order to explain how light, radio waves, X rays and other forms of radiation travel across space, physicists had proposed a universal medium call the ether , a kind of infinitely refined substance with infinite tensile strength. Light rays were the undulations of this ether. The host of fantastic, even contradictory properties which the ether was required to possess grew beyond bounds. until the ether hypothesis was eliminated by special relativity.
Heat : The internal consistency of Thermodynamics has always been somewhat problematic . Although a statistical theory, its predictions are deterministic : heat always flows from a warmer to a colder body. Before becoming celebrated through the discovery of relativity, Einstein published 24 papers on aspects of this subject. There is also a lower limit to heat loss: it had been observed that it was impossible to reduce any substance to Absolute Zero, a state of complete absence of motion. This phenomenon was explained by Quantum Theory and became incorporated into Nernst’s Law, otherwise known as the Third Law of
Thermodynamics .
(III) Obstacles were encountered in attempts to understand the phenomenon known as blackbody radiation . All substances when heated or set on fire change color: A poker in a brazier of burning coals will turn red, then blue, then white. The mechanism whereby heat converts to radiation raised difficulties ( even today there are problems associated with it ) ; two formulae had been devised to calculate the relationship between the intensity of the applied heat and the frequency of the emitted light. The first, the Rayleigh-Jeans Law, gave the correct predictions for low frequencies, while the other, Wien’s Law, worked for very high frequencies.
Max Planck discovered that both laws could be derived from the same expression if the assumption were made that, for any given frequency n , radiant energy was not released continuously but in discrete units proportional to n . That is to say, E = Energy = N x n x h , where
N is the number of units,
n is the frequency , and
h is the constant of proportionality, now known as Planck’s constant, given by h = 6.626 x 10-27 erg secs .
Einstein’s general theory of relativity was developed during World War I. Conceptually it was of such astounding difficulty for the times, so radical in its incorporation of advanced mathematical techniques, that he suffered a complete nervous collapse soon after its publication. Fortunately he recovered and lived on another 37 years. It is unusual insofar as it did not arise out of any anomaly or defect in prevailing theories. Instead it predicted things that no one had ever thought of looking for : the bending of light rays in the neighborhood of the sun, the slowing down of clocks in a gravitational field, a redshift in the light from distant stars produced by gravitational lensing , and others. The one unsolved problem solved by General Relativity dropped out by accident : a correction to Newtonian theory that accounted a precession of 43 seconds of arc per century in the long axis (perihelion ) of the orbit of Mercury .
Each of these new theories fashioned its primitive notions in terms of its own requirements, with no concern for the framework of the others. What was true then is still so today : Bring Relativity and Quantum Theory together and the harsh music coming from this combination is forbidding.
Since a statement of this sort may call anathema down upon me from professional physicists , a bit of space will be given over to clarifying it:
TIME in relativity theory ( both special and general) is treated as if it were exactly like any other dimension of spatial geometry . Velocity is interpreted as a rotation of time in the direction of space, or , as one should properly say, through space-time .
TIME in quantum theory, is treated as a parameter , while the spatial coordinates are treated as operators . This means that when an equation from classical physics is ‘reinterpreted’ or quantized at the subatomic level, length is replaced by a differential form , which is put into a special equation known as the Schrödinger wave equation , or simply ‘the wave equation’ . Time , on the other hand, undergoes no change. Both time and mass are treated as parameters, functioning essentially as constants of proportionality.
MOMENTUM in quantum mechanics is an irreducible primitive notion of the theory .
MOMENTUM in relativity is carried over unchanged from the older Newtonian theory and is defined as the product of two other primitive notions, ‘mass’ and ‘velocity’. Note that since time is treated spatially in relativity, velocity is dimensionless. This means that momentum is measured in units of mass alone.
What we have been learning is that , since 1900, every fundamental magnitude of physics has been redefined, not once but several times, and in different ways in different theories. It is because of this phenomenon , unique to the science of the modern world , that scientists are grappling with problems that had been by custom assigned to philosophy since approximately 100 B.S. [13] , the age of Heraclitus, Parmenides, Zeno , Pythagorus and Democritus. Then physics was a sub-division of philosophy, a second cousin to Metaphysics , Both physics and metaphysics became branches of theology at around 800 A.S. By the year 2000 , the two disciplines were fairly widely separated; Isaac Newton could smirk at the confusion of Bishop Berkeley over the calculus. A 23rd century physicist did not have to concern himself with the quarrels of Hume and Kant over the existence of cause and effect. . Little difference did it make to Maxwell what Hegel thought.
Deep foundational questions have been on the scientist’s menu since the early 1900’s A.C.E. Quantum theory emerged at the beginning of the century. After a break of about two decades it attained to its first synthesis with the Copenhagen Interpretation of 1926 . What began as one theory is now many: Relativistic Quantum Mechanics; Quantum Electrodynamics; Quantum Field Theory; Topological Quantum Field Theory; Axiomatic Quantum Theory; Quantum Thermodynamics; Quantum Chromodynamics; Quantum Gravity....
Only a small part of this vast subject can be touched on in the available space . In trying to get people interested in earlier versions of this book, I’ve often been told : “I’ve already read a book on that subject. I know all about quantum theory.” My reply has been , “You know much more than I do.” The field is enormous. For persons with enough background , it is advisable that they consult or study the standard textbooks, some of which are listed in the bibliography. There is still considerable value in a book like this one , which can be thought of as a kind of aerial reconnaissance photograph that gives an accurate , usable picture of the regional topography , without claiming to identify plant life, minerals or species of trees.
The Unknowable
The quantum theory is the most illuminating example of a particular philosophical perspective that has come to dominate all 20th century science: the acknowledge of the existence of phenomena, events, and causal connections which, by the structure of our relationship to nature, can never be known. Modern science is unique in history by the extent to which the unknowable has been elevated to a status comparable to that of the unknown .
We will be discussing the Copenhagen Interpretation, although we may find ourselves more interested in the rifts in this synthesis which opened up even during its formation : the debates between Max Planck, Werner Heisenberg , Max Born , Albert Einstein, Niels Bohr , and Erwin Schrödinger as to its completeness , credibility, fidelity to nature , and internal consistency . The divergence in perspective between Bohr and Einstein is of particular significance : it led, in the 80’s , to the discovery of non-locality . Though we will not be able to learn much about the numerous other branches of the theory , they will be referred to for examples as the occasion arises.
Quantum Theory Essentials
The phrases quantum theory, quantum mechanics, wave mechanics, and matrix mechanics , though signifying differing approaches , are synonyms . Quantum mechanics enters into the description of all phenomena at the atomic level : atoms, the strong and weak nuclear forces, the electromagnetic force , the elementary particles, light and radiation. There is as of yet no satisfactory theory of quantum gravity, a subject associated with Black Holes, the origins of the cosmos and the unification of all the forces of nature.
Paradoxes abound : every particle, the electron, proton , quark, and so forth, can also be interpreted as a wave. Every form of radiation can be interpreted as a particle. It was known by the 17th century that light beams behaves sometimes like wavefronts , sometimes like streams of particles. The mixing of colors, putting yellow and blue together to produce green, can only be explained by a wave model. Blackbody radiation is best understood thinking of light rays as particle streams . The principle whereby all physical entities behave like waves or particles depending on how one observes them, is known as Complementarity or The Principle of Complementary Images . It was first propounded by Niels Bohr at the physics conference in Como, Italy in September, 1927. It went through several revisions over the course of his career.
There are neither certain locations nor momenta . Accuracy in the determination of either one of these magnitudes is always at the expense of accuracy in the other. An absolute determination of position would cause an infinite uncertainty in the momentum . It would also need an infinite amount of energy to make, so we can forget about it. For the same reasons, we can’t hope for an absolute determination of the momentum.
These peculiarities of the quantum cosmos disappear, or fall below the threshold of perception , when the scale of our observations is at the level of day-to-day existence . This article of faith, which still encourages lively debate, is known as the Correspondance Principle and was also stated by Niels Bohr, in 1918.
Many would agree that the most controversial feature of quantum mechanics lies in the fact that its fundamental quantity, the range of values of the Schrödinger wave function y , corresponds to nothing one can measure in the observable universe. One might think of it as a kind of catalyst that, instead of going into some chemical reaction, gets put into an equation. Yet, from the form of the solutions to this equation we obtain all that can possibly be known about the behavior of a system at the atomic level. It therefore establishes the boundaries of unknowability . The wave function y does not itself correspond to a wave in the real world, but in an abstract mathematical construction known as phase space , or configuration space . To increase the confusion to the uninitiate, the form of the wave equation is elaborated in another mathematical construction, an infinite dimensional space known as Hilbert space . Fortunately the details need not concern us, save for one thing : its values are complex numbers, that is to say, they include a term in the square root of minus one, or i : y = y1 + iy2 .